Area of Circles Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version of this book, as well as other interactive content, visit www.ck12.org CK-12 Foundation is a non-profit organization with a mission to reduce the cost of textbook materials for the K-12 market both in the U.S. and worldwide. Using an open-content, web-based collaborative model termed the FlexBook®, CK-12 intends to pioneer the generation and distribution of high-quality educational content that will serve both as core text as well as provide an adaptive environment for learning, powered through the FlexBook Platform®. Copyright © 2013 CK-12 Foundation, www.ck12.org The names “CK-12” and “CK12” and associated logos and the terms “FlexBook®” and “FlexBook Platform®” (collectively “CK-12 Marks”) are trademarks and service marks of CK-12 Foundation and are protected by federal, state, and international laws. Any form of reproduction of this book in any format or medium, in whole or in sections must include the referral attribution link http://www.ck12.org/saythanks (placed in a visible location) in addition to the following terms. Except as otherwise noted, all CK-12 Content (including CK-12 Curriculum Material) is made available to Users in accordance with the Creative Commons Attribution/Non- Commercial/Share Alike 3.0 Unported (CC BY-NC-SA) License (http://creativecommons.org/licenses/by-nc-sa/3.0/), as amended and updated by Creative Commons from time to time (the “CC License”), which is incorporated herein by this reference. Complete terms can be found at http://www.ck12.org/terms. Printed: April 26, 2013 www.ck12.org Concept 1. Area of Circles CONCEPT 1 Area of Circles Introduction The On Deck Pads Do you remember Miguel? He had just finished working on figuring out the circumference of three different on deck pads for the pitchers to use while they warm up. Lets look at his dilemma again before we look at the area of the pads. Miguels latest task is to measure some different on deck pads for the pitchers to practice with. An on deck pad is a circular pad that is made up of a sponge and some fake grass. Pitchers practice their warm-ups by standing on them. They work on stretching and get ready to pitch the ball prior to their turn on the mound. Miguel has three different on deck pads that he is working with. The coach has asked him to measure each one and find the circumference and the area of each. Miguel knows that the circumference is the distance around the edge of the circle. He decides to start with figuring out the circumference of each circle. He measures the distance across each one. The first one measures 4 ft. across. The second measures 5 ft. across. The third one measures 6 ft across. Miguel has completed the first part of this assignment. He knows the circumference of each pad. Now he has to figure out the area of each. Miguel isnt sure how to do this. He cant remember how to find the area of a circle. Miguel needs some help. This is where you come in use this lesson to help you learn how to find the area of a circle. When finished, well come back to this problem and you can help Miguel figure out the area of each on deck pad. What You Will Learn By the end of this lesson, you will be able to demonstrate the following skills. 1 www.ck12.org • Recognize the formula for the area of a circle. • Find the area of circles given radius or diameter. • Find radius or diameter given area. • Find areas of combined figures involving parts of circles. Teaching Time I. Recognize the Formula for the Area of a Circle In the last lesson you learned how to calculate the circumference of a circle. Lets take a few minutes to review the terms associated with circles. A circle is a set of connected points equidistant from a center point. The diameter is the distance across the center of the circle and the radius is the distance from the center of the circle to the edge. We also know that the number pi, p, is the ratio of the diameter to the circumference. We use 3.14 to represent pi in operations. What does all of this have to do with area? Well, to find the area of a figure, we need to figure out the measurement of the space contained inside a two- dimensional figure. This is the measurement area. This is also the measurement inside a circle. You learned how to find the circumference of a circle, now lets look at using these parts to find the area of the circle. How do we find the area of a circle? The area of a circle is found by taking the measurement of the radius, squaring it and multiplying it by pi. Here is the formula. A = pr2 Write this formula down in your notebook. II. Find the Area of Circles Given the Radius or Diameter We already know that the symbol p represents the number 3.14, so all we need to know to find the area of a circle is its radius. We simply put this number into the formula in place of r and solve for the area, A. Lets try it out. Example What is the area of the circle below? 2 www.ck12.org Concept 1. Area of Circles We know that the radius of the circle is 12 centimeters. We put this number into the formula and solve for A. A = pr2 A = p(12)2 A = 144 p A = 452:16 cm2 Remember that squaring a number is the same as multiplying it by itself. The area of a circle with a radius of 12 centimeters is 452.16 square centimeters. Example Some students have formed a circle to play dodge ball. The radius of the circle is 21 feet. What is the area of their dodge ball circle? 3 www.ck12.org The dodge ball court forms a circle, so we can use the formula to find its area. We know that the radius of the circle is 21 feet, so lets put this into the formula and solve for area, A. A = pr2 A = p(21)2 A = 441 p A = 1;384:74 ft2 Notice that a circle with a larger radius of 21 feet has a much larger area: 1,384.74 square feet. Sometimes, you will be given a problem with the diameter and not the radius. When this happens, you can divide the measurement of the diameter by two and then use the formula. Example Find the area of a circle with a diameter of 10 in. First, we divide the measurement in half to find the radius. 10 ÷ 2 = 5 in Now we use the formula. A = pr2 A = 3:14(52) A = 3:14(25) A = 78:5 square inches This is our answer. 9N. Lesson Exercises Find the area of each circle. 1. Radius = 9 inches 2. Radius = 11 inches 3. Diameter = 8 ft. 4 www.ck12.org Concept 1. Area of Circles Take a few minutes to check your work. III. Find the Radius or Diameter Given the Area We have seen that when we are given the radius or the diameter of a circle, we can find its area. We can also use the formula to find the radius or diameter if we know the area. Lets see how this works. Example The area of a circle is 113.04 square inches. What is its radius? This time we know the area and we need to find the radius. We can put the number for area into the formula and use it to solve for the radius, r. A = pr2 113:04 = pr2 113:04 ÷ p = r2 36 = r2 p 36 = r 6 in: = r Lets look at what we did to solve this. To solve this problem we needed to isolate the variable r. First, we divided both sides by p, or 3.14. Then, to remove the exponent, we took the square root of both sides. A square root is a number that, when multiplied by itself, gives the number shown. We know that 6 is the square root of 36 because 6 × 6 = 36. The radius of a circle with an area of 113.04 square inches is 6 inches. Example What is the diameter of a circle whose area is 379:94 cm2? What is this problem asking us to find? We need to find the diameter (not the radius!). What information is given in the problem? We know the area. Therefore we can use the formula to solve for the radius, r. Once we know the radius, we can find the diameter. Lets give it a try. A = pr2 379:94 = pr2 379:94 ÷ p = r2 121 = r2 p 121 = r 11 cm = r The radius of a circle with an area of 379.94 square centimeters is 11 centimeters. 5 www.ck12.org Remember, this problem asked us to find the diameter, so were not done yet. How can we find the diameter? The diameter is always twice the length of the radius, so the diameter of this circle is 11 × 2 = 22 centimeters. As we have seen, we can use the area formula whenever we are given information about a circle. If we know the diameter or radius, we can solve for the area, A.